Approximations of Rockafellians, Lagrangians, and Dual Functions

Solutions of an optimization problem are sensitive to changes caused by approximations or parametric perturbations, especially in the nonconvex setting. This paper investigates the ability of substitute problems, constructed from Rockafellian functions, to provide robustness against such approximations. Unlike classical stability analysis focused on local changes around (local) minimizers, we employ epi-convergence to examine whether the approximating problems suitably approach the actual one globally. We show that under natural assumptions the substitute problems can be well-behaved in the sense of epi-convergence even though the actual one is not. We further quantify the rates of convergence that often lead to Lipschitz-kind stability properties for the substitute problems.